Landau-Ginzburg theories of non-Abelian quantum Hall states from non-Abelian bosonization

It is an important open problem to understand the landscape of non-Abelian fractional quantum Hall phases which can be obtained starting from physically motivated theories of Abelian composite particles. We show that progress on this problem can be made using recently proposed non-Abelian bosonization dualities in $2+1$ dimensions, which morally relate $U{\left(N\right)}_k$ and $\mathrm{SU}{\left(k\right)}_{-N}$ Chern-Simons-matter theories. The advantage of these dualities is that regions of the phase diagram which may be obscure on one side of the duality can be accessed by condensing local operators on the other side. Starting from parent Abelian states, we use this approach to construct Landau-Ginzburg theories of non-Abelian states through a pairing mechanism. In particular, we obtain the bosonic Read-Rezayi sequence at fillings $\nu =k/(kM+2)$ by starting from $k$ layers of bosons at $\nu =1/2$ with $M$ Abelian fluxes attached. The Read-Rezayi states arise when $k$ clusters of the dual non-Abelian bosons condense. We extend this construction by showing that $N_f$-component generalizations of the Halperin (2,2,1) bosonic states have dual descriptions in terms of $\mathrm{SU}{(N_f+1)}_1$ Chern-Simons-matter theories, revealing an emergent global symmetry in the process. Clustering $k$ layers of these theories yields a non-Abelian $\mathrm{SU}\left(N_f\right)$-singlet state at filling $\nu =k{N}_f/(N_f+1+kM{N}_f)$.

Figure 1

A schematic of our construction of LG theories for the RR states. $k$ copies of the $\nu =\frac{1}{2}$ Laughlin state coupled to scalars (left) are dual to $k$ copies of $\mathrm{SU}{\left(2\right)}_1$ coupled to scalars (right). The $\mathrm{SU}{\left(2\right)}_k$ Read-Rezayi states are obtained in the dual, non-Abelian language via pairing of the layers, represented by double-headed arrows. In the original Abelian theory, these correspond to nonlocal, monopole interactions.